## NEW! Section 6.4 – Work

### Instructions

- The first set of videos below explain the concepts in this section.
- This page also includes exercises that you should attempt to solve yourself. You can check your answers and watch the videos explaining how to solve the exercises.
- You can find additional practice problems for this section on the Practice Problems page for this Module.

### Concepts

- Using integration to calculate work
- Using Hooke’s Law to find the work done when stretching a spring and other application problems involving work and springs
- Finding the work done lifting a rope with a weight at the end
- Finding the work to pump water out of a tank

### Exercises

**Directions:** You should try to solve each problem first, and then click "Reveal Answer" to check your answer.
You can click "Watch Video" if you need help with a problem.

1. How much work is done in raising a 60 kg barbell from the floor to a height of 2 meters?

2. How much work is done in lifting a 50 pound weight from a height of 6 inches to a height of 18 inches?

3. When a particle is at a distance \(x\) meters from the origin, a force of \(f(x)=2x^2+1\) Newtons acts on the object, thus causing the object to move horizontally along the \(x\)-axis. How much work is done in moving the object from \(x=1\) to \(x=2?\)

4. A spring has a natural length of 1 m. If a 25-N force is required to keep it stretched to a length of 3 m, how much work is done in stretching the spring from 2 m to 5 m?

5. Suppose the work required to stretch a spring 1 foot beyond its natural length is 12 foot pounds.

- How much work is needed to stretch it 9 inches beyond its natural length?
- How far beyond its natural length will a force of 16 pounds keep it stretched?

- \(\dfrac{27}{4}\) ft-lbs
- \(\dfrac{2}{3}\) ft

6. A tank is in the shape of an upright cylinder with radius 5 feet and height 20 feet. The tank is half filled with water. Find the work required to pump the water to the top of the tank. **Clearly indicate where you are placing the axis and which direction is positive**.

7. A rectangular tank 10 m long, 8 m wide and 12 m is deep filled with water. Find the work required to pump the top 5 m of water out the top of the tank. **Clearly indicate where you are placing the axis and which direction is positive.**

8. A trough is in the shape of a half drum (half a cylinder lying on its side). The length of the tank is 10 meters and the radius is 4 meters. Assuming it is full of water, find the work done in pumping the water to the top of the tank. **Clearly indicate where you are placing the axis and which direction is positive.**

9. A triangular trough has a length of 5 feet, a distance of 2 feet across the top and a height of 3 ft. Assuming it is full of water, set up but do not evaluate an integral that gives the work done in pumping the water through a spout located at the top of the trough with length 0.5 feet. **Clearly indicate where you are placing the axis and which direction is positive.**

10. A tank is in the shape of a cone with radius \(r=3\) feet and height \(h=8\) feet. Assuming it is full of water, set up but do not evaluate an integral that gives the work it takes to pump the water out of the tank. **Clearly indicate where you are placing the axis and which direction is positive.**

11. A 100 foot long rope that weighs \(\displaystyle{{1}\over{5}}\) pounds per foot hangs vertically from the top of a tall building. Find the work done in pulling the entire rope to the top of the building.

12. A 500 foot rope that weighs 3 pounds per foot is used to lift an 80 pound weight up the side of a building that is 625 feet tall. Find the work done.

13. A 200 pound cable is 100 feet long and hangs vertically from the top of a tall building. How much work is done in pulling the first 10 feet of the cable to the top of the building?